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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\ Empirical Models for Tropical Storm \\ Windspeeds After Landfall}



\markboth{Landfalling Tropical Cyclones}{Landfalling Tropical Cyclones}

\author{Terrance Pendleton\\
Drake University\\
Des Moines  IA 50265 USA\\
terrance.pendleton@drake.edu
}

\makeStitlePDFLaTex


\begin{abstract}
We model the decay of tropical cyclone winds once a storm makes landfall. We use data from two recent storms from the National Hurricane Center to estimate parameters emanating from a differential equation using a first order exponential decay model. This modeling scenario is suitable for students who have experience with separable equations. 
\end{abstract}


\section*{SCENARIO DESCRIPTION}

\subsection*{Hurricane Forecasting Models} 

Due to the potential tragic nature of tropical systems, there is a need for the scientific understanding and modeling of these complicated phenomena in order to reduce unwanted destruction and prevent unnecessary deaths. Hurricanes are large, swirling storms with winds of 119 kilometers per hour (74 mph) or higher and are usually characterized by a low-pressure center, a closed low-level atmospheric circulation, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain. Coastal damage may be caused by strong winds and rain, high waves (due to winds), storm surges (due to severe pressure changes), and the potential of spawning tornadoes. Tropical storms also draw in air from a large area—which can be a vast area for the most severe storms—and concentrate the precipitation of the water content in that air (made up from atmospheric moisture and moisture evaporated from water) into a much smaller area. This continual replacement of moisture-bearing air by new moisture-bearing air after its moisture has fallen as rain, may cause extremely heavy rain and river flooding up to 25 miles inland from the coastline, far beyond the amount of water that the local atmosphere holds at any one time.

\vspace{1pc}
The National Hurricane Center (NHC) uses forecast models (any objective tool used to generate a prediction of a future event) in order to predict and prepare for tropical storms. Specifically, they are used as a way of guiding the NHC in their preparation of official storm track and intensity forecasts. The forecast models can take on a variety of different forms of varying complexity which seek to provide predictions for the storm's track, intensity, and/or wind radii. These methods use a variety of mathematical (and statistical) techniques such as multiple regression, dynamical systems and logistic growth. Many of these models focus on one particular aspect of the storm. For instance, the LGEM, or \textit{logistic growth equation model} is a statistical intensity model, which uses ocean heat content to predict the intensity of a tropical storm. 


\vspace{1pc}

In an effort to better understand and predict the path and intensity of a land-falling tropical system, we propose the development of a model for predicting the maximum sustained wind speed of landfalling tropical cyclones. 

\subsection*{Problem 1} 
The goal is to come up with a mathematical model for predicting the maximum sustained wind speed (MSWS) of a land-falling tropical cyclone. To do so, we must consider some simplifying assumptions. What are some assumptions that we can make which will aid us in developing the model? Below are some considerations. Which assumptions/considerations do you believe are most important to the development of the model? 

\begin{itemize} 
\item sea temperatures 
\item dry air
\item vertical wind shear (i.e. variation in wind velocity occurring along a direction at right angles to the wind's direction and tending to exert a turning force.) 
\item size of storm 
\item location of storm 
\end{itemize} 

\subsection*{Problem 2} 
To build the simplest model possible, we only assume that upon making landfall, tropical cyclone winds decay rapidly and in proportion to the current strength of the cyclone. See \cite{Kerry} for more information. Let $v(t)$ (in miles per hour) denote the MSWS at time $t$ (in hours) after the hurricane has made landfall. With the assumption given at the beginning of this problem, develop a differential equation which models this decay. Let $\alpha$ denote the proportionality constant. 

\subsection*{Problem 3} 
Solve the differential equation in Problem 2. Let $v_0$ denote the initial MSWS when the tropical cyclone makes landfall (corresponding to $t=0$). Compute $\lim_{t \to \infty}v(t)$. What do you observe about your limit? Realistically, as time progresses, it has been observed that the MSWS for a tropical cyclone decreases to a nonzero limiting wind speed. Equipped with this knowledge, adjust your model in problem 2 so that $\lim_{t \to \infty} v(t) = v_b$, where $v_b$ denotes this limiting nonzero wind speed. 

\subsection*{Problem 4} 
The solution that you have obtained in Problem 3 is a two-parameter solution in the sense that $\alpha$ (the decay constant) and $v_b$, the limiting wind speed, must be estimated from data. To this extent, consider the following data obtained from the National Hurricane Center, \cite{Undergrd}, for Hurricane Irma, an extremely powerful and catastrophic Cape Verde hurricane, which made landfall in the continental United States over Florida in 2017. See Figure \ref{Irma} for Hurricane Irma's track. 

\begin{figure}[h!]
\centering
\includegraphics[width=0.75\linewidth]{Irmatrack.png}
\caption{The track of Hurricane Irma \cite{Irma}}
\label{Irma}
\end{figure}

\begin{table}[h!]
\begin{center}

\label{tab:table1}
\begin{tabular}{c|c||c|c} 
\makecell{\textbf{time after landfall} \\ \textbf{(in hours)}} & \textbf{MSWS (in mph)} & 
\makecell{\textbf{time after landfall} \\ \textbf{(in hours)}} & \textbf{MSWS (in mph)}\\
\hline
0 & 130 & 12 & 100\\
1 & 130 & 13 & 100\\
2 & 130 & 14 & 100\\
3 & 120 & 15 & 85\\
4 & 120 & 18 & 75\\
5 & 115 & 21 & 70\\ 
6 & 110 & 24 & 65\\
7 & 110 & 27 & 60\\
8 & 110 & 30 & 50\\
9 & 105 & 33 & 45\\
10 & 105 & 36 & 35\\
11 & 105 & 39 & 25\\
\end{tabular}

\caption{Irma's MSWS after making landfall in Florida. \cite{Undergrd}}

\end{center}
\end{table}

We wish to estimate $\alpha$ and $v_b$ from this data. Using Excel worksheet \textit{hurricane.xlsx} along with this exercise, we can obtain the scatter plot in Figure~\ref{fig:Fig2} of the wind speeds at different times after landfall. 

\begin{figure}[h!]
\centering
\includegraphics[width=0.75\linewidth]{Irmaplot.png}
\caption{\label{fig:Fig2}Scatter Plot for the wind speeds of Irma After Landfall}
\end{figure}
\teacher{\textit{Remark: There are several ways for doing the parameter estimation. I picked Excel because of its accessibility and its ease of use for students who do not have sufficient computing experience.}}
We will use the Solver package in Excel to find optimal values for $v_b$ and $\alpha$ that best predicts the MSWS of Hurricane Irma. To use Solver in Excel for non-linear curve fitting, begin by creating a spreadsheet similar to the one in Figure~\ref{fig:Fig3}.
\begin{figure}[h!]
\centering
\includegraphics[width=0.75\linewidth]{spreadsheet.png}
\caption{\label{fig:Fig3}Spreadsheet Format for using Excel's Solver}
\end{figure}

Your solution should have the form 

\begin{equation} 
v(t) = A + B*e^{-Ct}.
\end{equation} 

Using the data as a guide, try to determine values for $v_0$, $v_b$ and $\alpha$ so that the computed wind speeds match the data. What factors did you use to decide how to find these \textit{optimal} values? In Figure 3, column C is the predicted MSWS based upon the initial guesses of $v_0$, $v_b$ and $\alpha$. The formula entered into cell C2 will depend on your solution obtained in Problem 3. For instance, if the computed solution were 

$$v(t) = C + A(1-e^{Bt}),$$
we would enter the corresponding formula: 

$$\text{\$G\$3 + \$G\$1*(1-EXP(\$G\$2*A1))}$$
and copy into all of column C, beginning with cell C2. Column D is the square of the difference between the recorded MSWS (Column B) and the predicted MSWS (column C), called \textit{chi squared} ($\chi^2$). The following formula should be entered into cell D2
$:=\text{(B2-C2)}\wedge 2$ and copied into all of column D. Cell G4 is the sum of the chi squares values, i.e. $\sum_{i=1}^{n}{\chi_i^2}$ (i.e. G4 = SUM(D2:D25)).

One particular way of obtaining these optimal parameter values is to \textit{minimize} $\sum_{i=1}^{n}{\chi_i^2}$ since if the predicted values for the MSWS of Irma are very close to the experimental curve then the value for $\sum_{i=1}^{n}{\chi_i^2}$ will be small. Squared values are chosen to avoid the cancellation effect since the difference between the computed wind speed and actual wind speed could have either sign. To minimize $\sum_{i=1}^{n}{\chi_i^2}$, we will use the ``Solver" add-in in Excel to find the values of $v_b$ and $\alpha$ that result in the minimum value for $\sum_{i=1}^{n}{\chi_i^2}$. Note that $v_0$ does not have to be estimated. Why? The procedure for using Excel's Solver is as follows: 

\begin{enumerate} 
\item You can access Solver in one of two ways, depending on which version of Excel is being used. Under the ``Tools" menu select ``Solver". A new pop-up window will appear. \textit{Remark}: If you do not see this as an option, the add-in will need to be installed. To access Solver, select ``Add-Ins" under the ``Tools" menu and check the solver add-in. 

\vspace{1pc}

Otherwise, you can access Solver in the Analysis group under the ``Data" tab. \textit{Remark}: If you do not see this as an option, the add-in will need to be installed. To access Solver, go to File $>$ Options. Click Add-Ins, and then in the Manage box, select Excel Add-ins. Click go. In the Add-Ins available box, select the Solver Add-in check box, and then click OK. After you load the Solver Add-in, the Solver command is available in the Analysis group on the Data tab. 


\item In the box labeled ``Set Target Cell", type in \$G\$4. (This is the sum that we are trying to minimize.)
\item Below this, select ``Equal To" the ``min" function since we are trying to minimize the value in cell G4. 
\item In the box labeled ``By Changing Cells" type \$G\$2:\$G\$3. This allows the solver to try and vary the values for the limiting wind speed $v_b$ and the decay constant $\alpha$ (which should be in cell G2 and G3) to minimize $\sum_{i=1}^{n}{\chi_i^2}$.
\item Now click on ``Solve". The program will alter your initial values to best fit the data. 
\item A new pop-up window will appear asking if you want to keep the new values or revert to your original values. Select ``keep solver solution" and click the ``OK" button. This gives the optimal values for $v_0$ and $\alpha$. 
\end{enumerate}

\subsection*{Problem 5} 

If you plotted the solution to the ODE obtained in Problem $3$ with the optimal values of $A$ obtained in Problem 4, you should get a graph that resembles Figure \ref{pred}. 

\begin{figure}[h!]
\centering
\includegraphics[width=0.75\linewidth]{MSWSIrma.png}
\caption{Predicted MSWS vs. Actual MSWS for Irma}
\label{pred}
\end{figure}

Using your graph as a guide, explain how your results verify the following observed claim regarding tropical storms: \teacher{\cite{hurrpaper}}

\vspace{1pc} 

\textit{Tropical storms whose circulations are partially over water (close to the coastline) decay less rapidly than those that are entirely over land.}

\vspace{1pc}
Why do you think this is the case? How could you adjust your model to account for this observation? 

\subsection*{Problem 6} 

During the 2018 hurricane season, Hurricane Michael roared ashore near the Florida Panhandle with a maximum sustained wind speed of 155 mph. Hurricane Michael formed in the Gulf of Mexico and made its way northwards until it made landfall near Panama City, FL. This made Hurricane Michael one of the strongest storms to ever hit the United States--and the first Category 5 storm to hit the United States since Hurricane Andrew in 1992. See Figure \ref{Mike} for Hurricane Michael's track. Consider the data in Table 2, obtained from \cite{Undergrd2}, which gives the MSWS for Hurricane Michael:



\begin{figure}[h!]
\centering
\includegraphics[width=0.75\linewidth]{Michaeltrack.png}
\caption{The track of Hurricane Michael \cite{Mike}}
\label{Mike}
\end{figure}

\begin{table}[h!]
\begin{center}

\label{tab:table2}
\begin{tabular}{c|c||c|c} 
\makecell{\textbf{time after landfall} \\ \textbf{(in hours)}} & \textbf{MSWS (in mph)} & \makecell{\textbf{time after landfall} \\ \textbf{(in hours)}} & \textbf{MSWS (in mph)} \\
\hline
0 & 160 & 10.5 & 70\\
0.5 & 155 & 13.5 & 60\\
1.5 & 150 & 16.5 & 50\\
2.5 & 140 & 19.5 & 50\\
3.5 & 125 & 22.5 & 50\\
4.5 & 115 & 25.5 & 50\\
5.5 & 100 & 28.5 & 50\\
6.5 & 90 & 31.5 & 50\\
7.5 & 85 & 34.5 & 50\\
8.5 & 80 & 37.5 & 60\\
9.5 & 75 & 40.5 & 65\\
\end{tabular}

\caption{Michael's MSWS after making landfall in Florida. \cite{Undergrd2}}
\end{center}
\end{table}

Using the previous problems as a guide, develop a mathematical model which determines the maximum sustained wind speed (MSWS) for Hurricane Michael as a function of time. Construct a plot similar to Figure \ref{pred}. Discuss your results. Is there an issue of ``overpredicting" (predicted wind speeds higher than the actual wind speed) and/or ``underpredicting" (as observed with the Irma model)? What possible adjustments could be made to this model to increase its accuracy?  




\begin{thebibliography}{98}

\bibitem{hurrpaper} John Kaplan and Mark DeMaria.  1995. A Simple Empirical Model for Predicting the Decay of Tropical Cyclone Winds after Landfall. {\it Journal of Applied Meteorology\/}. 34:  2499-2512.

\bibitem{Kerry} Emanuel, Kerry. 2005. {\it Divine Wind: The History and Science of Hurricanes\/}. New York: Oxford University Press.

\bibitem{Undergrd} Maximum Sustained Wind Speeds for Hurricane Irma. {\it Weather Underground \/} TWC Product and Technology, \url{https://www.wunderground.com/hurricane/atlantic/2017/hurricane-irma?map=history}. Accessed 31 March 2019

\bibitem{Undergrd2} Maximum Sustained Wind Speeds for Hurricane Michael. {\it Weather Underground \/} TWC Product and Technology, \url{https://www.wunderground.com/hurricane/atlantic/2018/Post-Tropical-Cyclone-Michael}. Accessed 18 July 2019

\bibitem{Mike} John L. Beven II, Robbie Berg, and Andrew Hagan ``Tropical Cyclone Report: Hurricane Michael." {\it National Hurricane Center \/} NOAA, \url{https://www.nhc.noaa.gov/data/tcr/AL142018_Michael.pdf}. Accessed 18 July 2019.

\bibitem{Irma} John P. Cangialosi, Andrew S. Latto, Robbie J. Berg. ``Hurricane Irma (AL112017) Tropical Cyclone Report.'' {\it National Hurricane Center\/}. NOAA. \url{https://www.nhc.noaa.gov/data/tcr/AL112017_Irma.pdf}. Accessed 18 July 2019.


\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}

This project grew out of an applied mathematics research group formed at Drake University in Fall 2017. At around this same time, Hurricane Irma was making news as one of the strongest, most devastating hurricanes to hit the United States since Hurricane Katrina. The students were interested in modeling the dynamics of a tropical storm system, and as a first step to realizing this goal, we came up with this simple model for predicting the decay of tropical cyclone wind speed's after landfall. The following semester, I taught a differential equations course and wrote this as a project for students after completing the unit on separable equations. 

Over the course of one class period (75 minutes), students completed Problems 1--Problems 4 in class in a group setting. I assigned Problems 5 and 6 as a homework assignment and asked each group to submit one complete report over Problems 1-6. 
%This is a set of COMMENTS about the scenario, for the teacher only. It might include a ``solution" or a number of solution strategies. However, the main narrative should be about how to use the scenario in teaching, e.g., what are prerequisites, how long might one devote to it, what resources for students could be expected to be used, what issues which cause students difficult or ease should be expected to come up, discussion of technology use, pointers to other files, etc. This set of COMMENTS might have its own pointers to data sets, supplementary files, videos, etc. for student and teacher use, the material for students could be something the teacher might use with discretion. There could be an extended bibliography \cite{Gause1934,Hutchinson1978a} for additional teacher resources, analyses, data, etc.


\vspace{2pc}
For \textbf{Problem 5}, I would like to mention two possible approaches for improving the accuracy of the model by correcting the overpredicting/underpredicting phenomenon. 
\textbf{Approach 1}: In \cite{hurrpaper}, the authors proposed using a \textit{correction term} in the model. The justification of this correction term comes from the following observations: 

\begin{enumerate} 
\item Hurricanes/Tropical storms whose circulations are partially over water decay less rapidly than those that are entirely over land. Thus, the correction term should require the calculation of the average distance inland D for each observation of wind speed. This observation is reflected in the underprediction (of MSWS of landfalling tropical storms)/overprediction observed in the results of Figure \ref{pred}. 
\item Prior research showed that 
\begin{itemize}
\item The mean radius of maximum wind speed of hurricanes is approximately 35 km.
\item The mean width of the eyewall of hurricanes is about 20--25 km. The eyewall of a hurricane is the roughly circular ring of deep convection which is the area of highest surface winds in a tropical cyclone.        
\item Slightly more than $50\%$ of the eyewall convection lies beyond the radius of maximum wind.
\end{itemize} 
\item On average, the eyewall convection surrounding the mean storm extends 50 km from the TC center. 
\item Research shows that when $D \leq 70$km, the eyewall of many storms remains partially over water, resulting in less decay than is observed for storms for which $D$ exceeds 70km. 
\end{enumerate} 

These observations motivated the need to add a correction term $C(t)$ to the model to include the effect of the distance inland. 

\vspace{1pc} 
\textbf{Approach 2}: Another approach is to use the pressure measurements associated with the storm to add robustness to the model. Taking advantage of the inverse relationship between the pressure and MSWS of a tropical cyclone (i.e. as the pressure increases, the MSWS decreases), one can develop a model for measuring the pressure of a tropical cyclone, call it $P(t)$, and then estimate the MSWS from the predicted value of $P(t)$ empirically. 

\vspace{1pc}
In Figure~\ref{fig:Fig6}, I have provided screenshots of the results obtained from using the data from Hurricane Michael for use in \textbf{Problem 6}.
\begin{figure}[h!]
\centering
\includegraphics[width=0.75\linewidth]{mikeopt.png}
\includegraphics[width=0.75\linewidth]{mikefit.png}
\caption{\label{fig:Fig6} Model Fit for Hurricane Michael (Problem 6)}
\end{figure}
}


%\begin{thebibliography}{99}




%\bibitem{Gause1934} Gause, G.~F. 1971. {\it The Struggle for Existence}. New York: Dover Publications, Inc. First published in 1934 by The Williams \& Wilkins Company and available completely on the world wide web at \url{http://www.ggause.com/Contgau.htm}. Accessed 31 March 2008.



%\bibitem{Hutchinson1978a} Hutchinson, G. Evelyn. 1978. {\it Introduction to Population Ecology\/}. New Haven CT: Yale University Press.

%\end{thebibliography}
%}





\end{document}

